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@Article{nowak-1992a, author = {Martin Nowak and Karl Sigmund}, title = {Tit for tat in heterogenous populations}, year = {1992}, volume = {355}, review-dates = {2005-02-21}, value = {ca}, month = {January}, pages = {250--253}, read-status = {reviewed}, journal = {Nature}, url = {http://www.ped.fas.harvard.edu/pdf_files/Nature92b.pdf}, hardcopy = {yes}, key = {nowak-1992a} }

$w$ is used as discount rate or the "constant probability for another round."

Axelrod noted: TFT is not the best strategy (for w > (T-R)/(T-P) = 2/4, there is no best strategy in the IPD.

Axelrod also noted: No single mutant can do better than TFT if w > max{(T-R)/(T-P), (T-R)/(R-S)} = 2/3.

Errors are a big problem for TFT. 1% error rate drops monomorphic TFT population effectiveness by 25% [Axelrod]

Introduce reactive strategies using triple (y,p,q), where:

- y is prob. that first move will be cooperate
- p is likelihood of cooperation next round if C from opponent last round
- q is likelihood of cooperation next round if C from opponent last round

Example mappings of strategies to triplet tuple:

- TFT = (1,1,0)
- AllD = (0,0,0)
- AllC = (1,1,1)
- STFT = (0,1,0) ... "suspicious tit-for-tat"

Framework actually more simple, since w=1, then y doesn't matter, and just (p,q) is necessary for tracking strategy differences for this paper.

Generous Tit-for-Tat (GTFT) is immune to less cooperative strategies and gives highest payoff. (nowak-1990a). Strategy is (1,q) with q=min{1-(T-R)/(R-S),(R-P)/(T-P)} = 1/3.

*Noted following in own experiments*:
"The strategies near AllD grow rapidly. TFT and all other
reciprocating strategies (near (1,0)) seem to have disappeared. But
an embattled minority remains and fights back. The tide turns when
'suckers' are so decimated that exploiters can no longer feed on
them."

Notes three-move memory, genetic algorithm learning experiments of Axelrod, which included noise. Optimal forgiveness in reactive strategies decreases if noise grows. (nowak-1990b, also supported by Axelrod and Dion's findings)

- axelrod-1984a
- see paper's bibliography

Problem Addressed:
Variation on: *How does cooperation emerge in IPD game?*

Main Claim and Evidence: Main claim: "...a small fraction of TFT players is essential for the emergence of reciprocation in a heterogenous population, but only paaves the way for a more generous strategy."

Assumptions:

- Stochastic, reactive players
- Infinite game (w=1)
- Straightforward $x_i^' = x_i f_i(x)/\bar{f} population updating scheme, proportional of fitness to average fitness.

Next Steps:

No detailed plan contained (very small article).

Remaining Open Questions:

Does forgiveness pay only under uncertainty... [but not too much or exploitation results]?

Contribution/Significance is good.

Quality of organization is outstanding.

Quality of writing is outstanding.

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